This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

learn more… | top users | synonyms (3)

0
votes
1answer
181 views

A reference and an explanation needed?

In my previous question I was asking for a method to construct a global point if we have local points with us which is here, but I got an answer, it didn't serve the entire purpose, but later on due ...
3
votes
0answers
367 views

algebraic ternary operators

In the analysis of certain Natural Language Processing queries, there is a certain algebra that arises with a ternary operator over a group with the following two properties: $$ ( ( a , b , c ) , d , ...
2
votes
0answers
111 views

About why Danielewski surfaces are the counterexamples for cancellation problem

The Danielewski surfaces $xy^n=1-z^2$ are the famous counterexamples for cancellation problem for affine spaces. I'm asking where to find the articles telling the whole story. Especially how to ...
2
votes
0answers
87 views

Introductory text about different stratification methods in higher-order logic and set theory

Could someone recommend me a good overview text about stratification of predicates, comprehension axioms, and other methods of avoiding the paradoxes in untyped or only loosely/relatively typed ...
4
votes
2answers
315 views

Does the $k$th forward difference of Radó's $\Sigma$ eventually dominate every computable function?

Let $\Sigma$ be Radó's Busy Beaver function, and let $\Delta[\Sigma]$ denote the forward difference of $\Sigma$, such that $\Delta[\Sigma] \ (n) = \Sigma(n+1) - \Sigma(n)$ for all $n \in \mathbb{N}$. ...
1
vote
2answers
372 views

What is the topological dual of $C_b(\mathbb{R})$

Consider the Banach space $C_b(\mathbb{R})$ of continuous bounded functions on $\mathbb{R}$ equipped with the sup-norm. 1) Do we know a precise description of its topological dual ...
3
votes
2answers
1k views

Books for high school students starting on college math

I am a student beginning high school from India. I have recently developed a taste for physics and mathematics. I am doing Lagrangian and Quantum Mechanics in Physics. But my mathematics is not too ...
1
vote
2answers
754 views

geometric meaning behind line integrals

What are some geometric meanings behind line integrals? I know if you have a curve on the xy plane and you are given a function $f(x,y)$ then the geometric meaning is a "curtain drawn" from the ...
0
votes
1answer
112 views

Advice for Area problem

I am very interested in a certain problem and I am wondering what methods currently exist to solve it. Given a curve defined by a function which maps any given arc length, s, from an arbitrarily ...
22
votes
2answers
715 views

Qual question archives?

Qual questions seem like a great way to study for a new topic, since they usually test slightly deeper understanding than typical questions in a textbook. Princeton has this great archive of questions ...
4
votes
0answers
260 views

Reference for Martingale version of Riesz representation theorem / Riemann–Stieltjes integral

(Please let me know if this is more appropriate as a MathOverflow question.) I can work out most of the following martingale generalization to the Riesz representation theorem and the ...
4
votes
2answers
121 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
3
votes
0answers
337 views

Definitions of weak (topological) mixing

Let $X$ be a compact (metric) space and $T:X\rightarrow X$ be a continuous map. Let $U_T:C(X)\rightarrow C(X)$ be the linear operator $U_T(f) = f\circ T$. Then Wikipedia (see ...
1
vote
3answers
119 views

Spectral Measures: References

I am trying to learn a little bit about the spectral theory of unbounded operators but the textbook we are using (Birman and Solomyak: Spectral theory of Self-Adjoint Operators in a Hilbert Space) is ...
5
votes
5answers
2k views

Music — Is the diatonic scale optimal in some sense?

I have recently found a mathematically-sound "proof" that the twelve-tone musical scale is optimal. I am looking for a similar explanation proving that the diatonic scale is optimal in some sense. ...
5
votes
3answers
490 views

Learning basic math?

Not sure if this is appropriate here but I am failing my calculus class and I basically have to take the last year of college over again. College I am trying to transfer to said I don't have the ...
17
votes
5answers
1k views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
2
votes
1answer
103 views

Free modules and the exactness of a sequence

When I read Thang Le's paper the coloured Jones polynomial and the A-polynomial of knots, it says in page 21 that: Since $R=\mathbb{C}[t^{\pm1}]$ is a PID, and $C$ is free over $R$. So if we tensor ...
2
votes
0answers
96 views

What can you do with rational solutions to linear equations?

I'm currently doing a project and for part of it I've been looking at rational solutions to linear eqautions in two vaiables. ie. ax+by=c. I'd like to add a bit about what we can use these types of ...
4
votes
2answers
1k views

First time passage decomposition for continuous time Markov chain

For discrete time finite Markov chain, the first passage time $T_j$ to visit state $j$, is determined from the recurrence equation: $$ p^{(n)}_{ij} = \sum_{k=0}^n f_{ij}^{(k)} p^{(n-k)}_{jj} = ...
1
vote
1answer
163 views

The fundamental gaussian identities of bayesian estimation

In bayesian estimation, when the model and plant noise is hold , the optimal estimator is Kalman filter. but I am wondering is there any literature that could prove the following gaussian identities? ...
1
vote
1answer
193 views

Every chain complex is quasi-isomorphic to a $\mathcal J$-complex

I found this in "Algebra & Topology" by Schapira, but I'm not able to prove it: Suppose $\mathcal J$ is a cogenerating family in an abelian category $\mathbf A$. Then for any positive complex ...
1
vote
0answers
69 views

What is the most efficient algorithm for constructing an irreducible polynomial?

Theorem: Assuming that the generalized Riemann hypothesis is true, there is a deterministic polynomial time algorithm to find an irreducible polynomial of degree $n$ over $\mathbb{F_p}$ The ...
2
votes
1answer
102 views

Asymptotic study of complex integrals

Anyone know a good reference for the asymptotic study of integrals of the form $\int_{\Gamma}f(s)e^{ng(s)}ds$ , $n\to\infty$, where $f(z)$ and $g(z)$ are analytic functions in the domain containing ...
4
votes
2answers
450 views

Linear programming for combinatorics/graph theory

I just went to a graph theory talk talking about various fractional graph parameters (but focusing on one). These were defined using linear programming. A question was asked, "How can we learn more ...
0
votes
2answers
598 views

Reference request: Calculus of Variations “cheat sheet”

I would appreciate any suggestions for "cheat sheets" (summary sheets) on the calculus of variations/ variational calculus in particular on the Euler -Lagrange equation, Lagrange multipliers, Legendre ...
4
votes
1answer
109 views

Locating a copy of a thesis

Does anyone have a transmittable copy of the thesis "Logical and cohomological aspects of the space of points of a topos" by Carsten Butz? The link provided in ...
5
votes
3answers
1k views

Linear combination of natural numbers with positive coefficients

I'm searching for a reference for the following result, so as to avoid writing a full proof in a paper. Alternatively, if a one-liner exists, I'd be glad to know it! Theorem: Let $a, b$ be two ...
26
votes
3answers
1k views

When did Fubini's name get applied to the theorem without measures?

Fubini's theorem, from 1907, expresses integration with respect to a product measure in terms of iterated integrals. The simpler version of this theorem for multiple Riemann integrals was used long ...
22
votes
9answers
2k views

Mathematics and Music

I have heard that, in recent years, many mathematicians as well as music theorists have applied different branches of mathematics to music. I would like to know about some books/resources relating to ...
9
votes
4answers
6k views

Difficulty level of Courant's book

I am currently studying Introduction to Calculus and Analysis by Richard Courant and Fritz John.I would like to compare Courant's book with Apostol's and Spivak's in terms of difficulty of the ...
15
votes
2answers
4k views

ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles ...
6
votes
1answer
171 views

Deformation of the Kähler structure on $CP^n$

Using the homogeneous coordinate on $CP^n$, we consider the open set $U_0:=\{[1, \ldots, z_n]\}$. Then the standard Kähler form of $CP^n$ can be written as $$ ...
2
votes
0answers
173 views

A full math history encyclopedia. Is there one?

Is there a book or a site or a periodic or a encyclopedia or something like this that's a complete reference in math history, talking about all known mathematicians and their achievements, not ...
7
votes
2answers
5k views

Rigorous Text in Multivariable Calculus and Linear Algebra

So I'm wanting a solid math book for Christmas. I have a solid background in Calculus and am currently working through baby Rudin. I really want a rigorous book dealing with multivariable calculus ...
13
votes
1answer
529 views

Polarization: etymology question

The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $$ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $$ where $Q(v) ...
8
votes
2answers
293 views

Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number ...
3
votes
3answers
425 views

Problems in elementary linear algebra

I'm looking for challenging problems in elementary linear algebra, i.e. mainly about matrices in the field of real numbers. Can anyone provide some references?
13
votes
2answers
1k views

Video lectures for Commutative Algebra

Are there any good video lectures for learning commutative algebra at level of Atiyah-Macdonald?
7
votes
1answer
466 views

Reference request: $L$-series and $\zeta$-functions

Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following: ...
9
votes
2answers
2k views

Category theory vs. Universal Algebra - Any References?

After seeing the answer to the question, "Category theory, a branch of abstract algebra", I would like to ask a question, Are there books/papers discussing the difference/indifference/comparison ...
3
votes
1answer
62 views

Eigenvalues of the matrix $(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$

$M_{[i],[j]}=(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$, where $1\le i_1<i_2<\cdots<i_k\le n$ and $1\le j_1<j_2<\cdots<j_k\le n$, can be taken to be an $\left(n\atop ...
3
votes
1answer
137 views

$n$-th homotopy group of subsets of $\mathbb{R}^n$

Let $X$ be a compact, path-connected subset of $\mathbb{R}^n$. I need a reference for the fact that the $n$-th homotopy group $\pi_n(X)$ is trivial. EDIT: Quite embarrassing. Indeed this is false, ...
2
votes
1answer
222 views

Category theory for describing systems?

In Rosetta Stone quantum description is interpreted from the category theory point of view. Systems (Hilbert spaces of wave functions) are objects and processes (linear operators) are arrows. But the ...
3
votes
1answer
223 views

Exact Constructions of Homotopy Fiber and Cofiber of Spectra

Given a map of spectra (pick whatever category you want), $f:X\to Y$, what are the exact constructions of the fiber and cofibers of this map? Does this depend in any deep way upon the category or ...
1
vote
1answer
228 views

Bessel's function

I read that $\int\limits_0^1 xJ_n(j_{na}x) J_n(j_{nb}x) dx={1\over 2}\delta_{ab}[J_n'(j_{na})]^2$, where $j_{na},j_{nb}$ are zeros of $J_n$, the Bessel function of the $n$th degree. Is there a ...
10
votes
5answers
2k views

Challenging problems in calculus

I'm looking for a textbook (or website etc.) which contains challenging problems in calculus. Problem in Real Analysis by Titu Andreescu is good, but slightly too advanced for me. Spivak's Calculus is ...
2
votes
1answer
85 views

Finding a group with a specific presentation

Is there a way to find the group whose presentation is given by $\langle a, b, c \mid a^{2} = b^{2} = c^{2} = abc = e, ab = ba, ac = ca, bc= cb\rangle$?
1
vote
1answer
97 views

Reference to a proof on simple function

On the Real Analysis - Modern Techniques and Their Application (second edition) by Gerald Folland, page 47 i found this theorem: "Let $f$ a measurable function. Then exists a sequence $(\phi_n)_{n \in ...
0
votes
1answer
75 views

reference for conditional expectation

Suppose $1\leq p<\infty$. Let $E$ be a Banach space. Consider a filtration $F_n$ on some probability space $\Omega$. Let $X\in L^p(\Omega,E)$ where $L^p(\Omega,E)$ denote the Bochner space. In ...